On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices
This paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>...
Saved in:
| Main Authors: | , , , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/23 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832589136703258624 |
|---|---|
| author | Abeer M. Albalahi Muhammad Rizwan Akhlaq A. Bhatti Ivan Gutman Akbar Ali Tariq Alraqad Hicham Saber |
| author_facet | Abeer M. Albalahi Muhammad Rizwan Akhlaq A. Bhatti Ivan Gutman Akbar Ali Tariq Alraqad Hicham Saber |
| author_sort | Abeer M. Albalahi |
| collection | DOAJ |
| description | This paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>E</mi><msub><mi>I</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula> and the general zeroth-order Randić index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula>. The minimum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>E</mi><msub><mi>I</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> in the class of all trees of fixed order containing no vertex of even degree are obtained for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>; also, the maximum value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> in the mentioned class is determined for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Moreover, in the family of all trees of fixed order and with a given number of vertices of even degrees, the extremum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>E</mi><msub><mi>I</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> are found for every real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∉</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. Furthermore, in the class of all trees of fixed order and with a given number of vertices of maximum degree, the minimum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>E</mi><msub><mi>I</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> are determined when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> does not belong to the closed interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>; in the same class, the maximum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> are also found for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The graphs that achieve the obtained extremal values are also determined. |
| format | Article |
| id | doaj-art-a598722551f34c96be85d0c31392cc37 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-a598722551f34c96be85d0c31392cc372025-01-24T13:22:10ZengMDPI AGAxioms2075-16802024-12-011412310.3390/axioms14010023On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree IndicesAbeer M. Albalahi0Muhammad Rizwan1Akhlaq A. Bhatti2Ivan Gutman3Akbar Ali4Tariq Alraqad5Hicham Saber6Department of Mathematics, College of Science, University of Ha’il, Ha’il P.O. Box 2240, Saudi ArabiaDepartment of Sciences and Humanities, National University of Computer and Emerging Sciences, B-Block, Faisal Town, Lahore 54770, PakistanDepartment of Sciences and Humanities, National University of Computer and Emerging Sciences, B-Block, Faisal Town, Lahore 54770, PakistanFaculty of Science, University of Kragujevac, 34000 Kragujevac, SerbiaDepartment of Mathematics, College of Science, University of Ha’il, Ha’il P.O. Box 2240, Saudi ArabiaDepartment of Mathematics, College of Science, University of Ha’il, Ha’il P.O. Box 2240, Saudi ArabiaDepartment of Mathematics, College of Science, University of Ha’il, Ha’il P.O. Box 2240, Saudi ArabiaThis paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>E</mi><msub><mi>I</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula> and the general zeroth-order Randić index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula>. The minimum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>E</mi><msub><mi>I</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> in the class of all trees of fixed order containing no vertex of even degree are obtained for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>; also, the maximum value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> in the mentioned class is determined for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Moreover, in the family of all trees of fixed order and with a given number of vertices of even degrees, the extremum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>E</mi><msub><mi>I</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> are found for every real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∉</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. Furthermore, in the class of all trees of fixed order and with a given number of vertices of maximum degree, the minimum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>E</mi><msub><mi>I</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> are determined when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> does not belong to the closed interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>; in the same class, the maximum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>R</mi><mi>α</mi><none></none><mprescripts></mprescripts><none></none><mn>0</mn></mmultiscripts></mrow></semantics></math></inline-formula> are also found for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The graphs that achieve the obtained extremal values are also determined.https://www.mdpi.com/2075-1680/14/1/23bond incident degree indicesvariable sum exdeg indexzeroth-order general Randić indexextremal valuestree |
| spellingShingle | Abeer M. Albalahi Muhammad Rizwan Akhlaq A. Bhatti Ivan Gutman Akbar Ali Tariq Alraqad Hicham Saber On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices Axioms bond incident degree indices variable sum exdeg index zeroth-order general Randić index extremal values tree |
| title | On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices |
| title_full | On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices |
| title_fullStr | On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices |
| title_full_unstemmed | On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices |
| title_short | On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices |
| title_sort | on trees with a given number of vertices of fixed degree and their two bond incident degree indices |
| topic | bond incident degree indices variable sum exdeg index zeroth-order general Randić index extremal values tree |
| url | https://www.mdpi.com/2075-1680/14/1/23 |
| work_keys_str_mv | AT abeermalbalahi ontreeswithagivennumberofverticesoffixeddegreeandtheirtwobondincidentdegreeindices AT muhammadrizwan ontreeswithagivennumberofverticesoffixeddegreeandtheirtwobondincidentdegreeindices AT akhlaqabhatti ontreeswithagivennumberofverticesoffixeddegreeandtheirtwobondincidentdegreeindices AT ivangutman ontreeswithagivennumberofverticesoffixeddegreeandtheirtwobondincidentdegreeindices AT akbarali ontreeswithagivennumberofverticesoffixeddegreeandtheirtwobondincidentdegreeindices AT tariqalraqad ontreeswithagivennumberofverticesoffixeddegreeandtheirtwobondincidentdegreeindices AT hichamsaber ontreeswithagivennumberofverticesoffixeddegreeandtheirtwobondincidentdegreeindices |